Another option might be array_remove(..., NULL) (introduced in 9.3) if tags.tag is NOT NULL (otherwise you might want to keep NULL values in the array, but in that case, you can’t distinguish between a single existing NULL tag and a NULL tag due to the LEFT JOIN):
SELECT objects.*,
array_remove(array_agg(tags.tag), NULL) AS tags,
FROM objects
LEFT JOIN taggings ON objects.id = taggings.object_id
LEFT JOIN tags ON tags.id = taggings.tag_id
If no tags are found, an empty array is returned.